Code notes below for teaching multivariable calculcus.

Definition: Tangent Plane

Let \(f: {\rm I\!R^2 \rightarrow\rm I\!R}\) be differentiable at \(\vec{x}_0=(x_0, y_0)\). The plane in \(\rm I\!R^3\) defined by the equation:

\[ z = f(x_0, y_0) + \left [ \frac{\partial f}{\partial x} (x, y_0)\right ](x - x_0) + \left [ \frac{\partial f}{\partial y}(x_0, y_0) \right ](y - y_0) \]

is called the tangent plane of the graph \(f\) at the point \((x_0, y_0)\).

Function Def.

library(plotly)
library(pracma)
library(Deriv)

plot_tangent_surface = function(f, x_0, y_0, grid_x = -10, grid_y = 10) {
  # Create meshgrid for plotting 
  m = meshgrid(grid_x:grid_y)
  nrows = nrow(m$X)
  X = as.vector(m$X)
  Y = as.vector(m$Y)

  # z point for surface.
  z_0 = f(x_0, y_0)

  # Compute partial derivatives. 
  df_dx = Deriv(f, 'x')
  df_dy = Deriv(f, 'y')
  
  # Create F surface
  F_MAT = f(X, Y)
  # Create tangent surface
  SURFACE = z_0 + df_dx(x_0, y_0)*(X - x_0) + df_dy(x_0, y_0)*(Y - y_0)
  
  # Convert back to matrix for plotly. 
  F_MAT = matrix(F_MAT, nrow = nrows)
  SURFACE = matrix(SURFACE, nrow = nrows)
  
  p <- plot_ly(z = ~F_MAT) %>% add_surface(
    contours = list(
        z = list(
        show=TRUE,
        usecolormap=TRUE,
        highlightcolor="#ff0000",
        project=list(z=TRUE)
      )
    )
  ) %>%
    layout(
      scene = list(
        camera=list(
          eye = list(x=1.87, y=0.88, z=-0.64)
        )
      )
    ) %>%
    add_surface(z = ~SURFACE)
  
  p
}

Running Code

# Define f
f = function(x, y) {
  x^2  + y^2
}

# Points of intercept with F. 
x_0 = -3
y_0 = -3

plot_tangent_surface(f, x_0, y_0, -15, 15)
# Define f
f = function(x, y) {
  x^(1/3) + y^(1/3)
}

# Points of intercept with F. 
x_0 = 1
y_0 = 1

plot_tangent_surface(f, x_0, y_0, -2, 5)